Advertisements
Advertisements
प्रश्न
Integrate the function in (sin-1x)2.
Advertisements
उत्तर
Let `I = int (sin^-1 x)^2 dx`
Put `sin^-1 x = theta`
⇒ x = sinθ
⇒ dx = cosθ dθ
∴ `I = int theta^2 cos theta d theta`
`= theta^2 int (cos theta) d theta - int (d/ (d theta) (theta^2) * int cos theta d theta) d theta`
`= theta^2 (sin theta) - int 2 theta (sin theta) d theta`
`= theta^2 sin theta - 2 int theta sin theta d theta + C`
`= theta^2 sin theta - 2 [theta * (- cos theta) - int 1 * (- cos theta) d theta] + C`
`= theta^2 sin theta + 2 theta cos theta - 2 int cos theta d theta + C`
`= theta^2 sin theta + 2 theta sqrt (1 - sin^2 theta) - 2 sin theta + C`
`= x (sin^-1 x)^2 + 2sin^-1 x sqrt (1 - x^2) - 2x + C`
APPEARS IN
संबंधित प्रश्न
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in x sin x.
Integrate the function in x sec2 x.
Integrate the function in x (log x)2.
`intx^2 e^(x^3) dx` equals:
Evaluate the following : `int e^(2x).cos 3x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : log (x2 + 1)
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int (sinx)/(1 + sin x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
Evaluate `int 1/(x(x - 1)) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
`∫ sin^(−1)` xdx is equal to ______.
Which expression is the integration-by-parts form for a product \(f(x)g(x)\)?
Evaluate \[\int x\cos x\,dx.\]
Let \[I=\int e^x\sin x\,dx.\] After applying integration by parts once, which equation is obtained?
